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JEE Main202325 Jan 2023Morning ShiftMathematicsHyperbolaActual

The vertices of a hyperbola H are ± 6 , 0 and its eccentricity is 5 2 . Let N be the normal to H at a point in the first quadrant and parallel to the line 2 x + y = 2 2 . If d is the length of the line segment of N between H and the y -axis then d 2 is equal to _____ .

Correct answer

0

Step-by-step solution

Given equation of hyperbola is H : x 2 36 - y 2 9 = 1 Any point on the hyperbola is P 6 sec θ , 3 tan θ . Equation of normal is 6 x cos θ + 3 y cot θ = 45 Also, this normal is parallel to the line 2 x + y = 2 2 , therefore slope of both lines must be equal , hence - 2 sin θ = - 2 ⇒ sin θ = 1 2 ⇒ θ = π 4 Equation of normal is 2 x + y = 15 And, this normal intersects y - axis at K , so P ≡ 6 2 , 3 and K ≡ 0 , 15 . So, P K 2 = d 2 = 6 2 - 0 2 + 3 - 15 2 &

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